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What is the sum of all the digits in all the integers from one to one million?

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How many solutions can you find to this sum? Each of the different letters stands for a different number.

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Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500?

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Try out some calculations. Are you surprised by the results?

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Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

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Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. . . .

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What happens when you add a three digit number to its reverse?

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Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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Where should you start, if you want to finish back where you started?

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Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E

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The Number Jumbler can always work out your chosen symbol. Can you work out how?

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Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

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Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

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In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?

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By selecting digits for an addition grid, what targets can you make?

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Explore the factors of the numbers which are written as 10101 in different number bases. Prove that the numbers 10201, 11011 and 10101 are composite in any base.

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The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

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Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

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32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.

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Find the values of the nine letters in the sum: FOOT + BALL = GAME

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Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

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How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?

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Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

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A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

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The number 27 is special because it is three times the sum of its digits 27 = 3 (2 + 7). Find some two digit numbers that are SEVEN times the sum of their digits (seven-up numbers)?

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Can you create a Latin Square from multiples of a six digit number?

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Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .

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This challenge is to make up YOUR OWN alphanumeric. Each letter represents a digit and where the same letter appears more than once it must represent the same digit each time.

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Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

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There are nasty versions of this dice game but we'll start with the nice ones...

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Explore the relationship between simple linear functions and their graphs.

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How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

Nowadays the calculator is very familiar to many of us. What did people do to save time working out more difficult problems before the calculator existed?

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Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your opponent.

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Suppose you had to begin the never ending task of writing out the natural numbers: 1, 2, 3, 4, 5.... and so on. What would be the 1000th digit you would write down.

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Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?

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A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

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Find the five distinct digits N, R, I, C and H in the following nomogram

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When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .

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Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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The number 3723(in base 10) is written as 123 in another base. What is that base?

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Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?

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115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?